Lax Familial Representability and Lax Generic Factorizations

WALKER, Charles Robert. Lax Familial Representability and Lax Generic Factorizations. Theory and Applications of Categories. New Brunswick: Mount Allison University, 2020, vol. 35, No 37, p. 1424-1475. ISSN 1201-561X.

Basic information

Original name

Lax Familial Representability and Lax Generic Factorizations

Authors

WALKER, Charles Robert (36 Australia, guarantor, belonging to the institution)

Edition

Theory and Applications of Categories, New Brunswick, Mount Allison University, 2020, 1201-561X

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Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Canada

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 0.545

RIV identification code

RIV/00216224:14310/20:00117741

Organization unit

Faculty of Science

UT WoS

000594117700037

Keywords in English

generic factorizations; lax conical colimit of representables

Tags

rivok

Tags

International impact, Reviewed

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Abstract

V originále

A classical result due to Diers shows that a copresheaf F: A -> Set on a category A is a coproduct of representables precisely when each connected component of F's category of elements has an initial object. Most often, this condition is imposed on a copresheaf of the form B (X, T-) for a functor T : A -> B, in which case this property says that T admits generic factorizations at X, or equivalently that T is familial at X. A classical result due to Diers shows that a copresheaf F: A -> Set on a category A is a coproduct of representables precisely when each connected component of F's category of elements has an initial object. Most often, this condition is imposed on a copresheaf of the form B (X, T-) for a functor T : A -> B, in which case this property says that T admits generic factorizations at X, or equivalently that T is familial at X. Here we generalize these results to the two-dimensional setting, replacing A with an arbitrary bicategory A, and Set with Cat. In this two-dimensional setting, simply asking that a pseudofunctor F: A -> Cat be a coproduct of representables is often too strong of a condition. Instead, we will only ask that F be a lax conical colimit of representables. This in turn allows for the weaker notion of lax generic factorizations (and lax familial representability) for pseudofunctors of bicategories T : A -> B. We also compare our lax familial pseudofunctors to Weber's familial 2-functors, finding our description is more general (not requiring a terminal object in A), though essentially equivalent when a terminal object does exist. Moreover, our description of lax generics allows for an equivalence between lax generic factorizations and lax familial representability. Finally, we characterize our lax familial pseudofunctors as right lax F-adjoints followed by locally discrete fibrations of bicategories, which in turn yields a simple definition of parametric right adjoint pseudofunctors. We also compare our lax familial pseudofunctors to Weber's familial 2-functors, finding our description is more general (not requiring a terminal object in A), though essentially equivalent when a terminal object does exist. Moreover, our description of lax generics allows for an equivalence between lax generic factorizations and lax familial representability. Finally, we characterize our lax familial pseudofunctors as right lax F-adjoints followed by locally discrete fibrations of bicategories, which in turn yields a simple definition of parametric right adjoint pseudofunctors.